Interface slip of steel–concrete composite beams reinforced with CFRP sheet under creep effect

Under the creep action of composite steel and concrete beams reinforced by carbon-fiber-reinforced polymer (CFRP) sheet, the face of the CFRP sheet, steel beam, and concrete slab beam produce relative slip. This slip affects the interface interaction, reduces the bearing capacity and stiffness of the members, and increases the deformation. In this paper, elastic and energy methods are used to analyze the interface forces between steel beams and concrete slabs reinforced by CFRP sheeting under the action of concrete creep. The calculation formulas for interface slip, axial force, and incremental deformation are established. The influence of design parameters on the mechanical properties of the interface is analyzed. Results show that the increments in interface slip, axial force, and deformation are zero on the 28th day. With increasing age, the increments in interface slip, axial force, and deformation gradually increase, and the increase is large in the first 100 days; it basically remains unchanged during the time interval from 100 to 1028 days. When the load increases by 5 N/mm (5 kN), the slip increments increase by approximately 0.004 mm, 0.002 mm, and 0.002 mm. The increments in axial force are approximately 19.4 kN, 15.9 kN, and 16.1 kN. The deformation increments increase by approximately 1.7 mm, 1.1 mm, and 0.6 mm.

h 0 + 250 ≤ 1500 ; ϕ 0 -nominal creep coefficient; f cm -average cubic compressive strength of concrete at age 28 days, f cm = 0.8f cu,k + 8 ; f cu,k -standard value of concrete cube compressive strength with 95% guarantee rate at age 28 days; β c (t − t 0 )-coefficient of development of the creep with time after loading; h-theoretical thickness of the member, where h = 2A/u ; A-member cross-sectional area; u-perimeter of the contact surface between the component and the atmosphere; RHannual average relative humidity of the environment; RH 0 = 100%; h 0 = 100 mm; t 1 = 1 day; f cmo = 10 Mpa; y si and y si (t)-respectively the vertical distance from the i-th layer of steel bars in the slab to the center of gravity of the slab converted section at time t 0 and time t ; y-vertical distance between the center of gravity of the beam and the slab at the moment of t 0 ; ε o -initial strain at the center of gravity of the plate at the moment of t 0 ; ϕ-initial curvature of the composite beam at time t 0 ; E s and E c -initial elastic moduli at time t 0 of the beam and plate, respectively; A s and A c -cross-sectional areas at t 0 of the initial beam and plate sections, respectively; I s and I c -moments of inertia at t 0 of the initial section ofbeam and plate, respectively; E si -the elastic modulus of the www.nature.com/scientificreports/ i-the layer of steel bars in the slab; A si -cross-sectional area of the i-th layer of steel bars in the slab; n-number of layers of reinforcement in the slab. Since the internal and external loads do not change during the t 0 − t time period, combined with the force on the section, the differential equation of the slip increment can be obtained.
The differential equation of the interface shear force increment can then be obtained according to the relationship between the interface shear force and the slip.
In t he for mu l a , 2 = n-the number of layers of steel bars in the plate; k L -stiffness of the connecting piece, k L = k/m ; k-stiffness of a single connecting piece; m-the longitudinal spacing of the connecting piece. The calculation formula of the interface slip increment under different loads can be obtained according to the boundary conditions. Uniform load action.
In the formula, Symmetric concentrated load action. Bending segment: Pure bend: Arbitrary concentrated load action. To the left of the loading point: To the right of the load point: Energy variation method. Based on the energy variational method theory 44 , it is assumed that there is no slip between the steel beam and the CFRP sheet. The displacement of the steel beam is U s1 , the displacement of the concrete is U c1 , the deformation of the beam is W 1 , and the displacement at the joint is � L1 = U s1 − U c1 +y(t)W ′ 1 .
As such, the potential energies of the structural system at the t moment are shown below: Strain energy at the joint: The total potential energy increment of the beam is shown in Eq. (13): According to the principle of minimum potential energy, the variation of Eq. (13) is further integrated stepby-step to obtain the following: Furthermore, δU s1 , δU c1 , δW are independent quantities, so Eq. (14) is integrated by division and reduced.
According to the balance of internal forces, the governing differential equation of the slip can then be obtained. Connection stiffness. The distribution curves of the slip increments with age under different stiffnesses are shown in Fig. 3. The interface slip increment decreases with increasing stiffness. The greater the stiffness, the smoother the slip increment curve. The amount of change to the interfacial slip increment decreases gradually with each increase in stiffness.

Incremental analysis of axial force
Incremental calculation of axial force. The differential equation of the axial force increment can be obtained from the relationship d�N(t) dx = �τ s (t) = k L �s(t) between the axial force increment and the slip increment.
The calculation formula of the axial force increment under different loads can be obtained according to the boundary conditions.

Uniform load action.
Symmetric concentrated load action. Bending segment: k L P e − l 0 /2 + e l 0 /2 e L + 1 e L− x − e x + k L P µ 1 α 2 − µ 2 β 2 2 α 2 (2x − L) + µ 2 βk L P e −αl 0 /2 + e αl 0 /2 2 α 2 − 2 α 3 e αL + 1 e αL−αx − e αx    Connection stiffness. The distribution curve of the axial force increment with age under different stiffness values is shown in Fig. 5. The axial force increment increases with increasing connection stiffness. The greater the load, the steeper the change in the curve of the axial force increment. Similarly, the greater the increase in the connection stiffness, the smaller the increase in the axial force increment.

Deformation incremental analysis of composite beams
Deformation increment calculation of composite beam. According to the relationship between deformation and curvature, combined with the differential equation of the slip increment, the differential equation of the deformation increment can be obtained. where .
According to the boundary conditions, the calculation formula of the deformation increment under different loads can be obtained.

Uniform load action.
Symmetric concentrated load action. Bending segment:

Pure bend:
Arbitrary concentrated load action. To the left of the loading point:    Connection stiffness. The distribution curve of the deformation increment with age under different stiffness conditions is shown in Fig. 7. In general, the deformation increment decreases with increasing connection stiffness, but the magnitude of the decrease is minimal, indicating that the connection stiffness is very important to the deformation increment of the component. This impact is not obvious.

Conclusion
In this study, elastic and energy methods are used to calculate the interfacial slip, axial force, and deformation increment in steel and concrete composite beams strengthened by CFRP sheeting under creep. The conclusions are summarized as follows: 1. The influence of the design parameters on the mechanical properties of the interface was analyzed. The calculation results show that the formula is correct and can be used to calculate the interface slip between the steel beam and the concrete slab reinforced by CFRP sheeting under the action of concrete creep. Based on the correct calculation formulas, the calculation formulas for interface slip, axial force, and deformation increment are derived. 2. Under the action of concrete creep, the slip amount, axial force, and deformation increment between the steel beam and the concrete slab strengthened by CFRP sheeting increase with increasing load. The greater the increase in the connection stiffness, the smaller the increase in the axial force increment. 3. The calculation results show that the increment of interface slip, axial force, and deformation are zero on the 28th day. As the age increases, the increment of interface slip, axial force, and deformation gradually increase; the increase is large in the first 100 days, and basically unchanged from 100 to 1028 days. 4. The calculation results also show that when the load is increased by 5 N/mm (5 kN), the slip increment increases by approximately 0.004 mm, 0.002 mm, and 0.002 mm, and the axial force increment increases by approximately 19.4 kN, 15.9 kN, and 16.1 kN. The deformation increment increases by approximately 1.7 mm, 1.1 mm, and 0.6 mm. 5. When the stiffness increases by one step, the change in the slip increment gradually decreases, and the axial force increases with an increase in connection stiffness. The larger the increase in connection stiffness, the smaller the increase in the axial force increment; the change in the deformation increment with increasing connection stiffness is minimal. 6. The theoretical derivation formula in this study is based on a series of assumptions and ignores the influence of some factors. Various factors should be considered in further research.

Data availability
Some or all data, models, or code generated or used during the study are available from the corresponding author by request.